Although it is a work of mathematics rather than metamathematics, it shows clearly by example how (usually) results about locales do not require the axiom of choice even when analogous results about topological spaces do. Paul Taylor has somewhat imprecisely written

In [Joh82] the public theorems about topology are marked with an asterisk, although the official meaning of that symbol is a dependence on the axiom of choice. (ASD I, page 3). Unfortunately for constructive mathematicians, excluded middle is not considered a form of choice.

Contents with links to nnLab pages

Besides the usual prefaces, bibliography, and indexes, there is a historical introduction, and each chapter concludes with notes on historical and metamathematical aspects. Otherwise, each of 7 chapters is divided into 4 sections, which in turn contain paragraphs that deal with essentially one idea each. For the moment, we list (with minimal processing) the definitions from the index in each section. There will also be some summaries of theorems; as in the book itself, an asterisk here indicates dependence on some form of choice beyond excluded middle (more precisely, a proof that cannot be internalised in an arbitrary boolean topos).